Can I Put 2D GIF Image on 3D Object
I am not sure if animated gifs are supported, but videos can be used as color inputs.
You can see more about video textures here:
https://www.blender.org/manual/render/blender_render/textures/types/video.html

in order to make the video appear on only one face of the box the best way would be to add one more material slot and assign a new material to that particular face.